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Experiment
K.-H. Rädler, M. Rheinhardt, E. Apstein, H. Fuchs
To cite this version:
K.-H. Rädler, M. Rheinhardt, E. Apstein, H. Fuchs. On the mean-field theory of the Karlsruhe
Dynamo Experiment. Nonlinear Processes in Geophysics, European Geosciences Union (EGU), 2002,
9 (3/4), pp.171-187. �hal-00302100�
Nonlinear Processes
in Geophysics
c
European Geophysical Society 2002
On the mean-field theory of the Karlsruhe Dynamo Experiment
K.-H. R¨adler, M. Rheinhardt, E. Apstein, and H. Fuchs
Astrophysical Institute Potsdam, An der Sternwarte 16, D-15482 Potsdam, Germany Received: 30 August 2001 – Accepted: 30 October 2001
Abstract. In the Forschungszentrum Karlsruhe an
experi-ment has been constructed which demonstrates a homoge-neous dynamo as is expected to exist in the Earth’s interior. This experiment is discussed within the framework of mean-field dynamo theory. The main predictions of this theory are explained and compared with the experimental results.
Key words. Dynamo, geodynamo, dynamo experiment,
mean-field dynamo theory, α–effect
1 Introduction
It is generally believed that the magnetic fields of the Earth, the Sun and other cosmic bodies result from dynamo mech-anisms. In the Forschungszentrum Karlsruhe a device has been constructed for an experiment which should demon-strate a homogeneous dynamo as is expected to exist in the Earth’s interior or in cosmic bodies (see, e.g. Stieglitz and M¨uller, 1996). The experiment was run the first time suc-cessfully in December 1999 (see M¨uller and Stieglitz, 2000, 2002; Stieglitz and M¨uller, 2001).
The basic idea of this experiment was proposed by Busse (1975, 1978, 1992). It is very similar to an idea discussed before by Gailitis (1967). The essential piece of the experi-mental device, the dynamo module, is a cylindrical container as shown in Fig. 1, with both radius and height somewhat less than 1 m, through which liquid sodium is driven by external pumps. By means of a system of channels with conducting walls, constituting 52 “spin-generators”, helical motions are organized. The flow pattern resembles one considered in the theoretical work of Roberts (1972) which proved to be capa-ble of dynamo action. It is sketched in Fig. 2.
It seems appropriate to discuss the experiment in the framework of the mean-field dynamo theory. Going beyond simple considerations of this kind (Busse, 1992; Busse et al., 1996, 1998; Stieglitz and M¨uller, 1996) a systematic theory has been developed with mean fields defined by averaging
Correspondence to: K.-H. R¨adler ([email protected])
over areas in planes perpendicular to the cylinder axis cov-ering the cross-sections of several cells (R¨adler et al., 1996, 1997a,b, 1998b). The essential induction effect of the fluid motion is then, with respect to the mean magnetic field, de-scribed as an anisotropic α-effect.
In order to obtain estimates of the self-excitation condition for the magnetic field in the experimental device and to give predictions of its geometrical structure, several kinematic mean-field dynamo models have been investigated, and cal-culations of the α-coefficient and related quantities have been carried out (R¨adler et al., 1996, 1997a,b, 1998b, 1999; R¨adler and Brandenburg, 2002). In addition, the back-reaction of the magnetic field on the motion has been taken into account in some approximation and so estimates for the saturation field strengths of the dynamo were derived (R¨adler et al., 1998a, 2000a,b). Parallel to the elaboration of the mean-field ap-proach to the theory of the experiment several direct numeri-cal simulations of the dynamo process have been carried out (Tilgner, 1996, 1997).
In this paper we give a summarizing representation of the mean-field theory of the experiment and compare the results with the measured data.
2 The mean-field concept
Let us assume that the magnetic flux density B inside the dynamo module is governed by the induction equation ∇ ×(η ∇ ×B − u × B) + ∂tB = 0 , ∇ ·B = 0 , (1)
where η is the magnetic diffusivity of the conducting fluid and u the velocity of its motion. The fluid is considered as incompressible, that is ∇ · u = 0.
We use a Cartesian co-ordinate system x, y, z as indicated in Fig. 1, with the z-axis aligned with the cylinder axis but z =0 in the middle of the dynamo module. The flow pattern inside the module is assumed to coincide, apart from some boundary layer, with a pattern as depicted in Fig. 2, showing periodicity in x and y with a period length 2a, and being independent of z.
Fig. 1. The dynamo module (after Stieglitz and M¨uller (1996)).
The signs + and − indicate that the fluid moves up or down, re-spectively, in a given spin generator. R = 0.85 m, H = 0.71 m,
a =0.21 m.
For the sake of simplicity we ignore until further notice (Sect. 7.1.2) the peculiarities at the curved boundaries of the dynamo module, that is, assume a flow pattern as in Fig. 2 for all x and y. We admit, however, at first a dependence of the flow on z. When speaking of a “cell” of this pattern we mean a unit like that defined by 0 ≤ x, y ≤ a. We further assume, again for simplicity, that η does not depend on x and y.
Let us follow the lines of the mean-field dynamo theory (see, e.g. Krause and R¨adler, 1980). For each given field F we define a mean field F by taking an average over an area corresponding to the cross-section of four cells in the xy-plane, F (x, y, z) = 1 4a2 a Z −a a Z −a F (x + ξ, y + η, z) dξ dη . (2)
We note that the applicability of the Reynolds averaging rules, which we use in the following, requires that F varies only weakly over distances a in x- or y-direction. By the way, all what follows applies also with a definition of F by averaging over an area corresponding to two cells only (Plu-nian and R¨adler, 2002), but we do not want to consider this possibility here in detail.
We split the magnetic flux density B and the fluid velocity u into mean fields B and u and remaining fields B0 and u0,
that is
B = B + B0, u = u + u0. (3)
Although in this paper B0 and u0 are more or less regular fields we will adopt the notation of mean-field theory and call them “fluctuations”. As long as we, in the sense ex-plained above, do not consider the situation near the curved boundaries we have u = 0, that is, u = u0.
Taking the average of Eq. (1) we see that B has to obey ∇ ×(η ∇ ×B − E) + ∂tB = 0 , ∇ ·B = 0 , (4)
Fig. 2. The Roberts flow pattern. The flow directions correspond to
the situation in the dynamo module if the co-ordinate system coin-cides with that in Fig. 1.
where E, defined by
E = u × B0, (5)
is a mean electromotive force due to the fluctuations u and B0.
The determination of E for a given u requires the knowl-edge of B0. Combining Eqs. (1) and (4) we easily arrive at ∇ ×(η ∇ ×B0−u × B − (u × B0)0) + ∂tB0=0 ,
∇ ·B0=0 , (6)
where (u × B0)0=u × B0−u × B0. We conclude from this that B0is, apart from initial and boundary conditions, deter-mined by u and B and is linear in B. We assume here that B0vanishes if B does so. Thus E, too, can be understood as
a quantity determined by u and B only and being linear and homogeneous in B. Of course, E at a given point in space and time depends not simply on u and B in this point but also on their behaviour in the neighbourhood of this point.
We adopt the assumption that B varies only weakly in space and time so that B and its first spatial derivatives in this point are sufficient to define the behaviour of B in the relevant surroundings. Then E can be represented in the form E = aijBj+bij k
∂Bj
∂xk
, (7)
where the tensors aij and bij kare averaged quantities
deter-mined by u. We use here and in the following the notation x1 = x, x2 = y, x3 = zand adopt the summation
con-vention. Of course, the neglect of contributions to E with higher-order spatial derivatives or with time derivatives of B remains to be checked in all applications (see Sect. 7.3).
Before giving results of calculations of E with specific as-sumptions on u, we write down its most general form com-patible with Eq. (7), which can be determined by standard methods of mean-field theory (see, e.g. Krause and R¨adler, 1980). Due to our definition of averages and the periodic-ity of the flow pattern, aij and bij k are independent of x and
y. Clearly a 90◦rotation of the flow pattern about the z-axis as well as a shift by the length a along the x-or the y-axis change only the sign of u so that simultaneous rotation and shift leave u unchanged. This is sufficient to conclude that aij and bij k are axisymmetric tensors with respect to the
z-axis. That is, aij is a linear combination of δij, εij lel and
eiej, and bij ka linear combination of εij k, δijek, δikej, δj kei,
εij lelek, εiklelej, εj klelei and eiejek. Here δij means the
Kronecker tensor, εij k the Levi-Civita tensor and e the unit
vector in z-direction. With this specification of aij and bij k
Eq. (7) turns into
E = −α⊥B − (αk−α⊥)(e · B) e − γ e × B
−β⊥∇ ×B − (βk−β⊥)e · (∇ × B) e
−β3e × ∇(e · B) + (e · ∇)B
−δ1∇(e · B) − δ2(e · ∇)B − δ3 e · ∇(e · B) e , (8)
with coefficients α⊥, αk, γ , β⊥, βk, . . ., which are averaged
quantities determined by u and are independent of x and y but may depend on z. The terms with α⊥and αkdescribe the
α-effect, which is in general anisotropic, those with β⊥and
βkgive rise to the introduction of a mean-field conductivity
different from the original electric conductivity of the fluid and again in general anisotropic. The term with γ describes a transport of mean magnetic flux like that due to a fluid mo-tion with the velocity −γ e. The remaining terms are less easily to interpret. We note that in contrast to the β⊥and βk
terms the β3term is not connected with ∇ × B but with the
symmetric part of the gradient tensor of B and can therefore not be interpreted in the sense of a mean-field conductivity.
We proceed now to the case in which u is independent of z (but return to the case in which it depends on z in Sect. 7.1.1). Consider for a moment B as a homogeneous field in the z-direction. Then ∇ × (u × B) vanishes, and we have to con-clude from Eq. (6) that B0 =0. This in turn leads to E = 0,
and therefore Eq. (8) can only be correct if αk = 0.
Re-turning again to arbitrary B we further consider the fact that averaged quantities determined by u can never imply a pos-sibility to distinguish between the positive and the negative z-directions. This means that E in the form given by Eq. (8) must be invariant under exchanging e with −e, which re-quires that γ = δ1=δ2=δ3=0. Thus we arrive at
E = −α⊥ B − (e · B) e
−β⊥∇ ×B − (βk−β⊥)e · (∇ × B) e
−β3e × ∇(e · B) + (e · ∇)B . (9)
Here the α-effect has an extremely anisotropic form. It is able to drive electric currents in the x- and y-direction but not in the z-direction.
3 Simple kinematic mean-field dynamo models
Let us consider simple kinematic mean-field dynamo mod-els which reflect essential features of the experimental de-vice. We assume here that the mean magnetic flux density B inside a cylindrical body which corresponds to the dynamo module is governed by Eq. (4). For the sake of simplicity we specify the electromotive force E so that it covers only the anisotropic α-effect and consider both η and α⊥as
indepen-dent of space coordinates and time. So we have inside this body
η ∇2B − α⊥∇ × B − (e · Be) − ∂tB = 0 ,
∇ ·B = 0 . (10)
In the outer space the α-effect is taken to be zero, and various assumptions concerning the electric conductivity are consid-ered, which will be specified later.
In models of that kind several types of magnetic fields showing different symmetries with respect to the axis and the middle plane of the cylinder are possible. Equations (10) al-low independent solutions B which vary like exp(imϕ) with the azimuthal co-ordinate ϕ of a cylindrical system whose axis coincides with that of the dynamo module. The fields with m = 0 are symmetric, such with m 6= 0 non-symmetric with respect to this axis. In the axisymmetric case there are again two independent types of fields. For one the poloidal part is antisymmetric and the toroidal part symmetric with respect to the middle plane, and for the other vice versa. Such fields are denoted by AS or SA, respectively. The poloidal part of an AS field is dipole-like, that of a SA field quadrupole-like. In the simplest non-axisymmetric case, that is m = 1, the field corresponds roughly to that of a dipole lying in the middle plane, but its field lines above and be-low this plane are distorted as it would happen with opposite rotations of the fluid in these regions about the z-axis. In the case m = 0 we have Bx = By = 0 on the z-axis, in
the case m = 1 we have Bz = 0, and for m ≥ 2 finally
Bx=By =Bz=0.
We may measure all lengths in units of the radius R of the cylindrical body considered and the time in units of R2/η. Then Eq. (10) takes the form
∇2B − C∇ × B − (e · B)e − ∂tB = 0 ,
∇ ·B = 0 , (11)
where C is a dimensionless measure of the α-effect, C = α⊥R
η . (12)
In general the solutions B of Eq. (11) for a given m are superpositions of independent solutions each of which varies with time like exp(pt ), where p is in general complex. For each such solution Eq. (11) together with proper boundary conditions pose an eigenvalue problem with p being the eigenvalue parameter. Of course, the eigenvalues p depend on C. Clearly the growth rate λ, given by λ = <(p), must be
Fig. 3. Concerning the numerical calculations: the cylindrical body
embedded in a sphere
negative for small C. For each type of solutions with a given m, in the case m = 0 with a given specification AS or SA, we define a marginal value C∗of C so that all λ are negative
for C < C∗, but at least one of them vanishes at C = C∗.
This marginal value C∗defines the self–excitation condition for the corresponding type of magnetic fields.
Estimates for the marginal values C∗where derived from models treated in other contexts in which the α-effect was not restricted to a finite cylinder but was assumed to act either in all space, in an infinite slab, in an infinite cylinder, or in a sphere. The results obtained in this way suggest that for our cylindrical body C∗ < 10 (R¨adler et al., 1996; Gailitis,
1967).
Several numerical studies of dynamo models as described above, that is, with the α-effect restricted to a finite cylinder, have been carried out. For most of them a code developed for spherical models (Fuchs et al., 1993) was used, with the cylinder embedded in an electrically conducting sphere sur-rounded by free space. As sketched in Fig. 3 the smallest sphere just containing the cylinder was chosen. The conduc-tivity of the parts of the sphere outside the cylinder was as-sumed to be equal to ξ times that inside the cylinder. In these calculations the ratio of radius R and height H of the cylin-der was fixed at R/H = 1. Using another method (Dobler and R¨adler, 1998), models with the same conductivity ev-erywhere inside and outside the cylinder and R/H = 1.21 were also investigated. We denote the two kinds of models by (i) and (ii). The marginal values C∗for some magnetic fields with low m are presented in Table 1 (see also R¨adler et al., 1996, 1998b). Figs. 4 and 5 exhibit examples of field structures. All these fields are steady, that is, non-oscillatory. Our results show that the non-axisymmetric field with m = 1 is clearly preferred over the axisymmetric and the other non-axisymmetric fields. That is, magnetic fields of the symmetry type m = 1 can be generated or maintained with the lowest requirements concerning the α-effect.
Results for a more sophisticated mean-field dynamo model will be given later (Sect. 7.4).
Table 1. Marginal values C∗ for cylindrical dynamo models of types (i) and (ii) and different types of magnetic fields
model m =0 m =0 m =1 m =2 AS SA (i) ξ = 1 8.22 8.46 6.41 8.62 (i) ξ = 0.01 8.64 9.18 7.70 9.67 (i) ξ = 0.001 9.02 9.60 8.12 10.12 (ii) 8.55 8.55 6.28 8.55
Fig. 4. Magnetic field of type m = 0 AS, ξ = 0.01. Left: isolines
of the toroidal part, right: field lines of the poloidal part
4 The α-effect under idealized conditions
4.1 General considerations
In order to formulate the self-excitation condition in terms of the rates of the flow through the spin generators we need to know how α⊥, or C, depends on them. In the following we
focus attention on the calculation of the coefficient α⊥in the
case in which u is independent of z (but will come to a case in which it depends on z in Sect. 7.1.1). For this purpose it is sufficient to restrict our considerations to the case in which B is a homogeneous field. For the sake of simplicity we further assume again that η is constant. Then u × B0is also constant, that is ∇ × (u × B0) =0, and Eq. (6) for B0takes the simple form
η ∇2B0+(B0· ∇)u − (u · ∇) B0−∂tB0= −(B · ∇) u ,
∇ ·B0=0 . (13)
We may assume that B0like B is independent of z. Let us put B0=B0⊥+B0kand u = u⊥+ukwith B0⊥=B
0−(e · B0)e
and B0k=(e · B0)e, and u⊥and ukdefined analogously. We
put further u⊥=u⊥u˜⊥and uk=uku˜k, where u⊥and ukare
factors independent of x and y characterizing the magnitudes of u⊥and uk, and ˜u⊥and ˜ukfields which are normalized in
some way. We may conclude from Eq. (13) that B0⊥depends only on u⊥ and not on uk, and that B0k depends again on
Fig. 5. Magnetic field of type m = 1, ξ = 0.01. Top: plane y = 0,
middle: plane x = 0, bottom: plane z = 0
implies that E and therefore α⊥ may depend in a complex
way on u⊥but must be linear and homogeneous in uk.
We will consider here two kinds of flow patterns, a highly idealized one previously investigated by Roberts (1972) and another one which is more realistic in view of the spin gener-ators of the experimental device. As indicated in Fig. 1 each spin generator consists of an central axial and an outer helical channel. In the ideal situation the fluid outside the channels is at rest.
For sufficiently small magnitudes of u the so-called second-order approximation can be justified, which consists
in the neglect of the two terms with u on the left-hand side of Eq. (13). In both cases we will start with this simple approx-imation but then proceed to results for arbitrary magnitudes of u.
4.2 Roberts flow
We define the Roberts flow by u = u⊥ a 2e × ∇χ − uk π 2 2 χe , χ =sin π ax sin π ay , (14)
or, more explicitly, ux= −u⊥ π 2 sin π ax cos π ay , uy= u⊥ π 2 cos π ax sin π ay , uz= −uk π 2 2 sin π ax sin π ay . (15)
Here u⊥is the average of the modulus of the velocity
com-ponent in the xy-plane perpendicular to a line running from the centre of a cell to its boundary taken over this line, e.g., the average of −uxat x = a/2 over 0 ≤ y ≤ a/2, or of uyat
y = a/2 over 0 ≤ x ≤ a/2, and ukis the average of the
mod-ulus of uzover the cross-section of a cell, e.g. 0 ≤ x, y ≤ a,
that is, u⊥= − 2 a a/2 Z 0 ux(a/2, y)dy , uk= − 1 a2 a Z 0 a Z 0 uz(x, y)dx dy . (16)
Using u⊥ and uk we define magnetic Reynolds numbers
Rm⊥and Rmkby Rm⊥= u⊥a 2η , Rmk= uka η . (17)
We also introduce volumetric flow rates V⊥and Vkby
V⊥ =
ah
2 u⊥, Vk = a
2u
k, (18)
where h means a length characterizing the pitches of the stream lines, which we will later identify with the pitch of the helical channel of a spin generator. We note that V⊥=Rm⊥hηand Vk=Rmkaη.
We consider first the second-order approximation, which applies in the limit of small u, more precisely for Rm⊥, Rmk 1. It allows us a simple determination of the
steady solution B0of Eq. (13) (see Appendix A). Calculating then u × B0we find α⊥= π2 32 a ηu⊥uk= π2 16 η aRm⊥Rmk= π2 16 V⊥Vk a2hη . (19)
Fig. 6. The functions φ (Rm⊥)and Rm⊥φ (Rm⊥)
According to our remarks in Sect. 4.1 the Relations (19) must also hold true for arbitrary uk, that is, arbitrary Rmk
and Vk. Therefore, generalizations of the form
α⊥= π2 32 a ηu⊥ukφ (u⊥a/2η) =π 2 16 η aRm⊥Rmkφ (Rm⊥) =π 2 16 V⊥Vk a2hη φ (V⊥/ hη) (20)
must apply for arbitrary u⊥and uk, or arbitrary Rm⊥, Rmk,
V⊥and Vk. Here φ is a function satisfying φ (0) = 1, which
remains to be determined.
Equations (13) for B0have been reduced to a system of or-dinary differential equations for its Fourier components with respect to x and y, and these have been integrated numeri-cally. From the result for B0 in the steady final state again u × B0and in the end α⊥have been calculated (see R¨adler
et al., 1997a, b). In this way the function φ (Rm⊥)shown in
Fig. 6 was determined. As can also be seen there α⊥,
con-sidered as function of Rm⊥, first grows with growing Rm⊥,
reaches a maximum at Rm⊥ =2.6, and then decays again.
This decay results from magnetic flux expulsion out of the ro-tating inner parts of each cell. By the way, in agreement with results of asymptotic studies (Soward, 1987) it was found that φ behaves like Rm−⊥3/2and, therefore, α⊥like Rm
−1/2
⊥
as Rm⊥→ ∞.
Let us interpret our result in view of an array of spin gener-ators. We denote the volumetric flow rates through the cen-tral and the helical channel of a spin generator by VC and
VH, respectively. Clearly, V⊥corresponds to VH, and Vk to
VC+VH. Then, using Eq. (20) and putting V⊥ = VH and
Vk=VC+VH, we arrive at α⊥= π2 16 VH a2hη(VC+VH) φ (VH/ hη) . (21)
Figure 7 shows the dependence of a2α⊥ on VCand VH for
h = 0.905a (chosen with a view to Eq. 32). There is a
Fig. 7. The dependence of a2α⊥on VCand VH, all three quantities
measured in units of aη, for h = 0.905a
value VC∗ (in the units of Fig. 7 we have VC∗ <2.5) so that α⊥(VC, VH)for any fixed VC < VC∗ grows with VH. For
VC > VC∗, however, α⊥(VC, VH grows for small VH only,
then reaches a maximum and decays again for larger VH.
This decay is again a consequence of the magnetic flux ex-pulsion from the inner parts of the cells. Later in Fig. 11 isolines of C in the VCVH-plane are shown, which because
of C = α⊥R/ηcan easily be interpreted as isolines of α⊥.
4.3 Spin generator flow
Let us now proceed to a flow pattern which is more realistic in view of the flow in the array of spin generators in the ex-perimental device. In order to describe the fluid velocity u we consider it at first only in a single cell, say 0 ≤ x, y ≤ a. We introduce there a cylindrical co-ordinate system %, ϕ, zwith the axis % = 0 at the centre of this cell, that is, at x = y = a/2. In this cell the fluid velocity u is, with respect to this co-ordinate system, assumed to be given by
u% =0 , uϕ =uϕ(%) , uz=uz(%) , (22)
with uϕ and uz depending on % only and vanishing for
% > a/2. The complete flow pattern in all xy-plane is then defined by periodic continuation of the pattern described for the cell considered here to all cells, with changes of the sign from each cell to its neighbouring cells as indicated in Fig. 2.
We specify u further by putting uϕ =0 , uz= −u(%)for 0 < % ≤ %1
uϕ = −ω(%)% , uz= −
h
2πω(%)for %1< % ≤ %2
uϕ =0 , uz=0 for % > %2. (23)
Here u and ω are arbitrary functions of %. Further %1and %2
mean the radius of the central channel and the outer radius of the helical one, respectively, and h the pitch of the helical channel; see Fig. 8. The − signs in Eq. (23) make that u and ωcan be considered as positive. The coupling between uϕ
Fig. 8. Cross-section of a spin generator
and uzin %1 < % ≤ %2considers the constraint on the flow
resulting from those walls of the helical channel which are no cylindrical surfaces. We define further the averages u⊥,
ukCand ukHof the relevant velocities,
u⊥= 2 a %2 Z %1 ω(%)% d% ukC= 2π a2 %1 Z 0 u(%)% d% , ukH= h a2 %2 Z %1 ω(%)% d% , (24)
and note that ukH = (h/2a)u⊥. On this basis we define
magnetic Reynolds numbers Rm⊥, RmkCand RmkHby
Rm⊥= u⊥a 2η , RmkC= ukCa η , RmkH= ukHa η , (25) where, of course, RmkH = (h/a)Rm⊥. Note that the
av-erages u⊥, ukCand ukHare related to the length a/2 or the
area a2and not to the actual extents of the respective flows. By this reason also Rm⊥, RmkCand RmkH have to be
in-terpreted with some care. Finally we introduce the volumet-ric flow rates VCand VH through the central and the helical
channel, VC=a2ukC, VH = ah 2 u⊥=a 2u kH. (26)
In the second-order approximation, that is for suf-ficiently small magnitudes of u, more precisely for Rm⊥, RmkC, RmkH 1, the quantity α⊥can be calculated
by taking the average of, say, (u × B0)x over a single cell
ignoring the contributions to B0resulting from the flow out-side, and dividing it by Bx. It can be shown that these
contri-butions to B0 produce only such parts of (u × B0)x which
vanish under averaging (see Appendix B). Assuming then that u in the considered cell is given by Eq. (22) and van-ishes outside, it is again easy to find the steady solution of
Eq. (13) (see Appendix A). So we arrive at
α⊥= π a2η %2 Z 0 uϕ(%) % Z 0 uz(%0)%0d%0 +uz(%)% %2 Z % uϕ(%0) d%0 d% . (27)
Interestingly enough, as can be shown by an integration by parts, the two double integrals on the right-hand side are equal to each other. We may therefore also write
α⊥= 2π a2η %2 Z 0 (uϕ(%) % Z 0 uz(%0) %0d%0) d% = 2π a2η %2 Z 0 (uz(%) % %2 Z % uϕ(%0) d%0) d% . (28)
Let us evaluate these relations for α⊥with the more
spe-cific assumptions (Eq. 23) on uϕand uz. We find then
α⊥= a 2ηu⊥ ukC+ 1 2ukH = η aRm⊥ RmkC+ 1 2RmkH = VH a2hη VC+ 1 2VH . (29)
Obviously the axial flow in the central channel of the spin generator, where no azimuthal flow exists, is more effective in view of α⊥ than the axial component of the flow in the
helical channel. Note that in particular the relation between α⊥, VCand VHapplies independently of %1and %2.
We leave now the second-order approximation. With the same arguments as used in the case of the Roberts flow we find that the general forms of α⊥, which apply for arbitrary
u⊥, ukC, ukH, Rm⊥, · · · VCand VH, are given by
α⊥= a 2ηu⊥ ukCφC(u⊥a/2η) + 1 2ukHφH(u⊥a/2η) = η aRm⊥ RmkCφC(Rm⊥) + 1 2RmkHφH(Rm⊥) = VH a2hη VCφC(VH/ hη) + 1 2VHφH(VH/ hη) . (30) The functions φC and φH, which have to satisfy
φC(0) = φH(0) = 1, may depend, apart from the arguments,
also on the profile of ω.
In order to determine α⊥and so φCand φHthe Eqs. (13)
have been solved numerically in the region −a ≤ x, y ≤ a using proper periodic boundary conditions (R¨adler and Bran-denburg, 2002). For the sake of simplicity both u and ω were taken as constants, that is, rigid-body motions of the fluid, or piston profiles, were assumed in each of the channels. Re-sults for φC(Rm⊥)and φH(Rm⊥)obtained in this way are
Fig. 9. The functions φC(Rm⊥)and φH(Rm⊥)for rigid-body
mo-tion of the fluid in each of the channels with %1=a/4, %2 =a/2
and h = 0.905a
shown in Fig. 9. The dependence of a2α⊥on VCand VHfor
%1=a/4, %2=a/2 and h = 0.905a (chosen in agreement
with Eq. 32) is represented in Fig. 10. 1 Again the com-ment given with Fig. 7 applies according to which there is a value V∗
C (here V
∗
C <4) so that α⊥(VC, VH)for any fixed
VC< VC∗grows with VH, for VC> VC∗, however, grows for
small VHonly, then reaches a maximum and decays again for
larger VH. We also refer to Fig. 12 which shows isolines of
C, which because of C = α⊥R/ηcan easily be interpreted
as isolines of α⊥.
5 The self-excitation condition of the experimental device in comparison with experimental results
5.1 Self-excitation condition
In the following we will apply the results obtained so far to the experimental device and compare them with experimen-tal findings. For this purpose we choose for the radius R and the height H of the dynamo module the values
R =0.85 m , H =0.71 m . (31)
More precisely, as indicated in Fig. 1 these values correspond to the “homogeneous part” of the dynamo module, which does not include the regions with connections between the spin generators, etc. We further adopt for the edge length a of a spin generator, the radius %1 of the inner channel, the
outer radius %2and the pitch h of the helical channel
a =0.21 m , %1=0.25a , %2=0.5a , h = 0.19 m . (32) 1The results presented in some of our earlier papers (R¨adler
et al., 1997a,b, 1998b) were obtained with an analytic solution of Eqs. (13) for a single spin generator ignoring the influence of the neighbouring ones, what is not completely correct beyond the second-order approximation. The numerical investigations men-tioned confirm the essential features of the results but show that corrections of numerical data are necessary if Rm⊥is no longer
small compared to unity. These corrections are considered here.
Fig. 10. The dependence of a2α⊥on VCand VH, all three quantities
measured in units of aη, with %1=a/4, %2=a/2 and h = 0.905a
Finally we put for the magnetic diffusivity of liquid sodium
η =0.1 m2/s . (33)
We return first to the dimensionless measure C of the α-effect introduced with Eq. (12) and express it by the volu-metric flow rates VC and VH. With the result (21), which
was obtained for the Roberts flow, we find the dependence of C on VC and VH depicted in Fig. 11. In the same way
the result (30) for the spin generator flow leads to the de-pendence shown in Fig. 12. For the Roberts flow we can show that, when admitting arbitrary VCand VH, each isoline
of C cuts the VH-axis and continues until infinite VC.
Pre-sumably the same applies to the spin generator flow. The non-uniqueness of VHas a function of VCis, of course, again
a consequence of the magnetic flux expulsion from the inner parts of the spin-generators. Remarkably enough, in the re-gions of VCand VHwhich are of interest for the experiment,
that is 0 < VC, VH <200 m3/h, the isolines of C essentially
coincide for both kinds of flow patterns. Considering the spin generator flow as more realistic than the Roberts flow we will refer to the isolines of C shown in Fig. 12 in what follows.
We recall that in our approach the self-excitation condi-tion for the dynamo reads C ≥ C∗ and that values of C∗ obtained under various assumptions are listed in Table 1. For any given value of C∗ we have a “neutral line” C = C∗ in
the VCVH-plane separating the region in which the dynamo
can work from the one where it can not. As can be seen from Figs. 11 and 12 dynamo action should be possible for arbitrarily small VH if only VCexceeds a sufficiently large
value depending on VH. Likewise a dynamo should work
with VC=0 and a sufficiently large VH.
5.2 Experimental results
Using data measured in the experiment (M¨uller and Stieglitz, private communication) the real neutral line in the VCVH
-plane separating dynamo and non-dynamo regions has been determined. Figure 13 shows a detail of Fig. 12 with this
Fig. 11. Isolines of C, obtained with the result (21) for the Roberts
flow, in the VCVH-plane. Both VCand VHin m3/h. When starting
from the numerical values related to the units used in Fig. 7 those related to m3/h follow by multiplication by a factor 75.6.
Fig. 12. Isolines of C, obtained with the result (30) for the spin
generator flow, in the VCVH-plane. Both VCand VHin m3/h. When
starting from the numerical values related to the units used in Fig. 10 those related to m3/h follow by multiplication by a factor 75.6.
line added. It corresponds to values of C∗ in the interval 8.4 · · · 9.3.
Magnetic field measurements have been carried out at sev-eral points along the axis of the dynamo module. It turned out that the field there consists mainly of x- and y-components. Compared to them no noticeable z-component was observed. This applies likewise to the components of B (see Appendix C) and indicates that, as expected, the generated fields are of the symmetry type m = 1. It should, however, be noted that the variation of the x- and y-components of B derived from the observed field along the axis of the dynamo mod-ule (see again Appendix C) are not in satisfactory agreement with calculated field structures as shown in Fig. 5.
The fact that the values of C∗derived from the
measure-Fig. 13. A detail of measure-Fig. 12 with the experimentally determined
neu-tral line (the thick line) separating regions with and without dynamo action
ments are somewhat higher than those given for m = 1 in Table 1 is well understandable. As we will explain in more detail below (Sect. 7) improvements to the simple models for which Table 1 applies lead to higher values of C∗. Such im-provements consider in particular the variability of the coef-ficient α⊥and the occurrence of effects described by αkand
γ near the boundaries of the dynamo-active body as well as the effects described by β⊥, βkand β3inside this body.
6 On the back-reaction of the magnetic field on the fluid flow and the saturation of the magnetic field
6.1 A simple dynamo model involving the back-reaction of the magnetic field
So far we considered kinematic dynamo models only, that is, we ignored the influence of the Lorentz forces on the fluid flows. The Lorentz forces are of second order in the magnetic field, and their influence on the fluid flow grows with the magnetic field and limits so its magnitude. In order to study this process in detail in addition to the induction equation (Eq. 1) the hydrodynamic equations involving the Lorentz forces have to be taken into account.
Instead of investigating the very complex problem which occurs in this way we deal here only with a simple model of the dynamo in the nonlinear regime (see also R¨adler et al., 1998a). It considers no other consequence of the back-reaction of the magnetic field on the fluid motion than the magnetic contribution to the pressure drops in the channels of the spin generators. Influences of the magnetic field on the flow profiles in the channels (as discussed in R¨adler et al., 2000a) or the generation of motions in the fluid outside the channels are ignored.
We start again from Eqs. (10) for B, with α⊥considered as
a function of the flow rates VCand VH, and add two equations
as consequences of the Navier-Stokes equation, relating the flow rates to the pressures built up by the pumps and the pres-sure losses due to the hydraulic resistance and the magnetic field. The full set of these equations reads
∂tB = η∇2B + α⊥(VC, VH) ∇ × B − (e · B)e ,
∇ ·B = 0,
dtVC=κC PC(VC) − RC(VC) − LC( ˜B, VC, VH),
dtVH=κH PH(VH) − RH(VH) − LH( ˜B, VC, VH) ,(34)
where, of course, the first line must be completed using proper boundary conditions. Here κCand κHare factors of
the structure s/ρml where s is the cross-section of the
con-sidered type of channels, ρmthe mass density of the fluid and
lthe total length of the considered circuit. PCand PHare the
pressures generated by the pumps in these circuits, RC and
RH the pressure drops due to the hydraulic resistances, LC
and LHthe pressure drops due to Lorentz forces, and ˜Bis a
quantity depending on the magnitude of the relevant compo-nents of the magnetic field, which will be specified later. We point out that according to our above assumption α⊥depends
only via VCand VHon the magnetic field, that is, a possible
dependence via the flow profiles is not taken into account. It should be noted that there is one circuit in the experi-mental device which contains the central channels of all 52 spin generators but there are two circuits for the helical chan-nels, each feeding 26 of them. Here these two circuits are assumed to be equal to each other, that is, described by one flow rate, VH, only.
We specify Eq. (34) by further assumptions concerning PC, PH, RC, RH, LC and LH. For the pressures generated
by the pumps we put
PC=kCPCo(1 − cP CVC) , PH=kHPHo(1 − cP HVH) , (35)
where the factors kCand kHdescribe with which fractions of
the maximum pressure the pumps work, 0 < kC, kH ≤ 1.
Further, PCoand PHoare the maximum pressures, and cP Cand
cP H are constants considering the pressure drops inside the
pumps under load (see Stieglitz and M¨uller, 1996). For the pressure losses due to the hydraulic resistance we assume RC=RCo 1 + cR C 1 + c 0 R C VC 1/4! VC2, RH=RHoVH2,(36)
where RoC, RHo, cR Cand cR C0 are constants (see again Stieglitz
and M¨uller, 1996). The main contributions to the resistances are due to the bends of the tubes.
Corresponding relations for LCand LH will be given
be-low.
6.2 Estimates of the Lorentz forces
For an estimate of the Lorentz forces we assume that B is a homogeneous field. Then the force exerted on a unit volume of the fluid, f , is given by
f = 1 µ0
(∇ ×B0) × (B + B0) , (37)
where µ0is the magnetic permeability of free space.
We restrict ourselves first to the second-order approxima-tion as explained in the context of Eq. (13) and replace in the same sense B + B0 in Eq. (37) simply by B. For the calculation of the averages of f which are of interest below it is then again justified to consider a single spin generator only, that is, to ignore any motion in the neighbouring ones. As in Sect. 4.3 we consider the spin-generator defined by 0 ≤ x, y ≤ a and use again the co-ordinate system %, ϕ, zintroduced there. We can easily find a steady solution B0 of Eq. (13) (see Appendix A) and calculate f according to Eq. (37). Averaging its ϕ and z-components over ϕ and de-noting these averages by ˆfϕ and ˆfzwe have
ˆ fϕ = − 1 2σ uϕB 2 ⊥, fˆz= − 1 2σ uzB 2 ⊥, (38)
where σ is the electric conductivity of the fluid, σ = 1/µ0η,
and B⊥the mean magnetic flux density in the xy-plane. Of
course, ˆfϕ and ˆfzdepend on % if uϕand uzdo so.
Consider first a central channel. The magnetic pressure drop per unit length (dpm/dl)Cof this channel is, apart from
the sign, just the average of ˆfzover its cross-section, that is
dpm dl C = 1 2σ huziB 2 ⊥, (39)
where huzimeans the average of uzover the volume or, what
is the same, over the cross-section of the channel. Denot-ing this cross-section by sC, where sC = π %21, and using
huzisC=a2ukCwe find further dpm dl C = σ a 2B2 ⊥ 2sC ukC= aB⊥2 2µ0sC RmkC= σ B⊥2 2sC VC. (40)
Consider next a helical channel. For the pressure drop per unit length (dpm/dl)Hwe can derive a relation analogous to
Eq. (39) with huzireplaced by cos δhuϕi +sin δhuzi. Here δ
means the angle between some central stream line at a radius % = % and a circle with % = % and z = const , that is, tan δ = h/2π %, and huϕiand huziare now averages over the
volume of the channel or, what is the same, over its section with a plane z = const . We define % byR%1%2uϕ(%) % d% =
%R%2
%1 uϕ(%) d% and put % = ξ 0(%
1+%2)/2 where ξ0 is a
factor close to unity. Further we introduce the cross-section sH as the area of a plane fitting into the channel and being
perpendicular to the central stream line mentioned, that is sH=(%2−%1) hcos δ. So we arrive at dpm dl H = σ ahξ 0B2 ⊥ 4sH u⊥ = hξ 0B2 ⊥ 2µ0sH Rm⊥= σ ξ0B⊥2 2sH VH. (41)
Let us now leave the second-order approximation in Eq. (13) and the analogous one in Eq. (37). Using ana-lytical solutions of Eq. (13) for an isolated spin generator, that is, ignoring as before the influences of the neighbouring
Fig. 14. The function ψC(VC, VH)for two special values of VC
given by the labels of the curves, with %1 = a/4 and %2 = a/2. Both VCand VHin m3/h. The function ψCvaries monotonically
with VC.
ones, and assuming rigid-body motions, that is piston pro-files, (dpm/dl)Cand (dpm/dl)Hfor both channels have been
determined for arbitrary VCand VH. We present the result in
the form dpm dl C = σ B 2 ⊥ 2sC VCψC(VC, VH) dpm dl H =σ ξ 0B2 ⊥ 2sH VHψH(VC, VH) , (42)
with two functions ψC and ψH satisfying ψC(VC,0) =
ψH(VC,0) = 1. These functions with %1 = a/4 and
%2=a/2 are shown in Figs. 14 and 15.
We complete now the Eqs. (34) to (36) by LC=B⊥2L˜C, L˜C=cL CVCψC(VC, VH)
LH=B⊥2L˜H, L˜H=cL HVHψH(VC, VH) (43)
where cL Cand cL Hare constants of the structure σ l/2s, with
σ, l and s being the electric conductivity of the fluid, the total length of all channels in the considered circuit and s their cross-section.
6.3 Saturated dynamo states
We consider now our dynamo model defined by the Eq. (34) together with Eqs. (35), (36) and (43) for a state in which B, VC and VH neither grow nor decay. We already know
from kinematic dynamo models that B for fixed VCand VH
shows a non-oscillatory behaviour, and we could not find any example of a different behaviour of B, VCand VHin the case
considered here. Therefore we restrict our attention here to the steady case.
Steady solutions of the equations for B in Eq. (34) require that C(VC, VH)takes its marginal value C∗. Hence the
con-sequences of the Eqs. (34) with (35), (36) and (43) for the steady case read
C(VC, VH) = C∗
Fig. 15. The function ψH(VC, VH)for two special values of VC
given by the labels of the curves, with %1 = a/4 and %2 = a/2.
Both VCand VHin m3/h. The function ψH varies monotonically
with VC.
PC(VC) − RC(VC) − B⊥2L˜C(VC, VH) =0
PH(VH) − RH(VH) − B⊥2L˜H(VC, VH) =0 . (44)
Eliminating B⊥2 from the last two lines of Eq. (44) we find PC(VC) − RC(VC)L˜H(VC, VH)
− PH(VH) − RH(VH)L˜C(VC, VH) =0 . (45)
If all other relevant parameters are given the first line of Eq. (44) together with Eq. (45) allows us to determine a pair, or possibly several pairs, of values VCand VH without
con-sidering B⊥2. With the help of the second or the third line of Eq. (44) we can afterwards find the corresponding value of B⊥2 . We must, however, discard all pairs of VCand VH for
which B⊥2 takes negative values.
On the basis of Eq. (44), completed by Eqs. (35), (36) and (43), we may calculate the quantities VC, VHand B⊥if, for
example, C∗, kCand kHare given. For this purpose we need
the numerical values of PCo, PHo, cP C, cP H, RCo, RHo, cR C,
c0R H, cL C and cL H. Without going into details we note that
the parameters of the device (see Stieglitz and M¨uller, 1996) lead to PCo=PHo=710 kPa , cP C=cP H =10.1 (m3/s)−1 RCo =1.31 · 108Pa (m3/s)−2, RoH=1.99 · 108Pa (m3/s)−2 cR C=3.54 · 10−1, cR C0 =3.54 · 10 −2m3/s cL C=1.88 · 1010kg(m4sT2)−1, cL H=2.31 · 1010kg(m4sT2)−1 (46)
(see also R¨adler et al., 2000a,b).
In Table 2 flow rates VCand VHand the quantity B⊥
char-acterizing the magnitude of the generated magnetic field in steady states of the dynamo are listed for various values of C∗, kCand kH. In addition, the total power N is given which
is needed to maintain these states as well as the relative frac-tion fohmfed into the magnetic field and converted into heat
Table 2. The flow rates VCand VH and the measure B⊥of the
magnitude of the magnetic field for steady states of the dynamo, further the total power N needed to maintain these steady states and its relative fraction fohmcorresponding to ohmic dissipation
C∗ kC kH VC VH B⊥ N fohm [m3/h] [m3/h] [10−4T ] [kW] 8.0 1 1 106 92 303 42 0.73 1 0.5 127 78 215 28 0.57 0.5 1 84 108 256 36 0.63 0.5 0.5 102 94 169 21 0.45 8.5 1 1 110 96 291 43 0.70 1 0.5 130 82 204 29 0.52 0.5 1 89 113 243 37 0.59 0.5 0.5 106 98 155 21 0.40 9.0 1 1 114 100 281 44 0.67 1 0.5 134 86 193 29 0.49 0.5 1 93 118 230 38 0.54 0.5 0.5 110 103 140 22 0.33 9.5 1 1 118 104 270 45 0.64 1 0.5 138 90 182 30 0.44 0.5 1 97 123 216 39 0.49 0.5 0.5 114 108 123 22 0.26
in Eq. (46) have noticeable uncertainties, the values of B⊥,
N and fohm must be considered as rough estimates. In this
sense they are in good agreement with the experimental re-sults.
7 Steps toward a refined theory of the experiment
7.1 Boundary effects
7.1.1 The plane bottom and top boundaries of the dynamo module
The calculations of the electromotive force E reported above ignored the fact that the fluid flow is restricted to the dy-namo module and that near its bottom and top covers there are flows between the spin generators. In order to get an idea on the influence of this type of boundary effect on the excita-tion condiexcita-tion of the dynamo, a calculaexcita-tion of the coefficients α⊥, αkand γ occurring in Eq. (8) has been carried out in the
second-order approximation no longer assuming a Roberts flow as given by Eq. (14) but the modified flow defined by u = u⊥ a 2e × ∇(f⊥χ ) + uk( a 2) 2∇ × e × ∇(f kχ ) , χ =sin(π ax)sin( π ay) , (47)
where f⊥and fkare functions of z (see R¨adler et al., 1996).
With f⊥=fk=1 we return to Eq. (14). We think, however,
of functions f⊥and fkwhich are equal to unity in some inner
part of the dynamo module only but decay with growing |z|
Fig. 16. A flow pattern with connecting flows between the spin
generators
and vanish outside the module. A flow pattern in a region with varying f⊥and fkis shown in Fig. 16. With this flow
α⊥is no longer independent of z, and αkand γ are unequal
to zero in and near to the regions with varying f⊥and fk.
Calculations of these coefficients have been carried out for the two cases in which there is either free space beyond the covers of the dynamo module or a medium at rest with the same electric conductivity as the fluid (R¨adler et al., 1996). We may represent the results in the form
α⊥= π2 32 a ηu⊥ukh⊥(ζ ) = π2 16 η aRm⊥Rmkh⊥(ζ ) αk= π2 32 a ηu⊥ukhk(ζ ) = π2 16 η aRm⊥Rmkhk(ζ ) γ = π 2 16 a ηu 2 kk(ζ ) = π2 16 η aRm 2 kk(ζ ) , (48)
with the dimensionless functions h⊥, hkand k of ζ = 2z/H .
If f⊥ and fkare symmetric in z then h⊥ and hk are again
symmetric but k is antisymmetric in ζ .
Let us consider the simple example in which f⊥ = 1 in
0 ≤ |ζ | ≤ 1, further fk=1 in 0 ≤ |ζ | ≤ 1 − , fk=p5(|ζ |)
in 1 − ≤ |ζ | ≤ 1, and f⊥ = fk = 0 for |ζ | ≥ 1, where
is a constant and p5a polynomial of the fifth degree such
that fkand its first and second derivatives are continuous
ev-erywhere. The profiles of h⊥, hkand k for the case of free
space beyond the covers of the dynamo module are shown in Fig. 17. Those for the case with a fluid at rest are very similar. Note that the sign of k corresponds to a transport of magnetic flux out of the dynamo module.
The influence of the connecting flows in the sense dis-cussed so far, that is, of the reduction of α⊥and the
Fig. 17. The functions h⊥, hkand k for f⊥and fkas described in
the text and = 0.4 in the case of free space beyond the covers of the dynamo module
self-excitation of the dynamo has been studied with a sim-ple model in which the dynamo acts in an infinite slab sur-rounded by free space (R¨adler et al., 1996). In this model the marginal value C∗, here related to a definition of C analo-gous to Eq. (12) but with the thickness of the slab instead of R, depends on as introduced above and on q = Rmk/Rm⊥,
too. Compared to the case with constant α⊥and vanishing αk
and γ , the value of C∗grows both with and q. For ≤ 0.2 and q ≤ 1 the increase is less than 10%. We may expect that in the experimental device the boundary effects discussed so far let C∗, compared to the idealized case, grow to a similar extent.
7.1.2 The curved boundary of the dynamo module It is difficult to determine the mean velocity u of the fluid or the mean electromotive force E for the curved boundary regions of the dynamo module. In any case u must deviate from zero, and it must vary with the azimuth ϕ with the pe-riod π . Likewise the coefficients of E as, e.g., α⊥must show
such a variation with ϕ. It is the neglect of these boundary effects which made that in the mean-field approach consid-ered so far the dynamo module appeared to be an axisym-metric object and, as a consequence, there was no coupling between B-fields differing in m. Of course, in a more de-tailed theory this axisymmetry of the dynamo module and its consequences must disappear. In the experiment indeed a clearly preferred direction for the generated fields occurs (M¨uller and Stieglitz, private communication).
7.2 Mean-field conductivity, etc.
So far we have not considered the contributions to the elec-tromotive force E which are connected with derivatives of B. Dealing now with these contributions, we again restrict ourselves for the sake of simplicity to the case in which the flow pattern is independent of z, that is, to the β⊥, βkand β3
terms in Eq. (9). As already mentioned (Sect. 2) the first two
can be interpreted in the sense that they contribute to a mean-field conductivity, and it is to be expected that they lead to an enhanced dissipation of the mean magnetic field. However, the last one does not need to act in this sense. A straightfor-ward calculation with the Roberts flow defined by Eq. (14) using the second-order approximation yields
β⊥= a2 64η(u 2 ⊥+ π2 4 u 2 k) = η 16(Rm 2 ⊥+ π2 16Rm 2 k) βk= a2 32ηu 2 ⊥= η 8Rm 2 ⊥ (49) β3= − a2 64η(u 2 ⊥− π2 4 u 2 k) = − η 16(Rm 2 ⊥− π2 16Rm 2 k)
(see R¨adler et al., 1996). Note that β⊥ and βkare positive
definite, that is, must indeed lead to an enhanced dissipation of the mean magnetic field, whereas β3may take both signs
so that it is difficult to predict its influence. Note also that we have β3=β⊥−βk.
The β⊥and βk-effects necessarily lead to higher values of
C∗for any given dynamo model. Estimates with a very sim-ple model show that this tendency is maintained if in addi-tion the β3-effect is taken into account (R¨adler et al., 1996).
Again an increase of C∗ up to 10% is to be expected as a
consequence of the effects discussed here. This statement is in agreement with results of another way of calculating the α⊥-effect and the β⊥, βkand β3-effects and their influences
on C∗(R¨adler and Brandenburg, 2002). 7.3 On the limits of the mean-field approach
As usual in mean-field dynamo theory we have adopted the assumption that B varies weakly in space and time so that all contributions to E with higher than first-order spatial deriva-tives and with any time derivaderiva-tives of B are negligible. We may consider the radius R of the dynamo module as a charac-teristic length scale of B and the edge length a of a spin gen-erator as the averaging length scale. According to Eqs. (31) and (32) we have a/R = 0.25. That is, the above assump-tion is not well satisfied and the statements derived from mean-field considerations should be checked in an indepen-dent way.
In this context investigations of subharmonic solutions of the original Roberts dynamo problem (Tilgner and Busse, 1995; Plunian and R¨adler, 2002) are of interest. We rely here on the recent one of them (Plunian and R¨adler, 2002), which is widely elaborated in view of the Karlsruhe dynamo, and adopt the definitions introduced above for the original z-independent Roberts flow. In particular such subharmonic solutions of the induction equation for B with u specified by Eq. (14) have been considered which possess no part in-dependent of x and y and whose period lengths in x- and y-direction are integer multiples of the length of a diagonal of a cell in the flow pattern, that is,
√
2N a with an integer N . An arbitrary period length in z-direction was admitted, here denoted by
√
2κa with an arbitrary positive real constant κ. Subharmonic fields B of that kind have been determined by
numerical solution of the eigenvalue problem posed by the Fourier-transformed induction equation.
Instead of the cylindrical dynamo module we consider now a rectangular “dynamo box” with the edge lengths L in the x- and y-direction and H in the z-direction. We consider a subharmonic field B such that the dynamo box contains just a “half wave” of its leading Fourier mode, that is, the mode with the largest period lengths. This means N a/
√
2 = L and κa/
√
2 = H . We then interpret the leading Fourier mode as the mean field B. Instead of characterizing the situation considered by the parameters N and κ we may also use the aspect ratios L/H and a/H of the dynamo box and of the spin generators.
Before presenting specific results derived from subhar-monic solutions of the Roberts dynamo problem let us have a look on a result of the mean-field approach, that is, a solution of the Eq. (10) for B with α⊥given by Eq. (20) which fits in
the same sense to our dynamo box as we required it above for the leading mode of a subharmonic field B. As can be easily shown the self-excitation condition reads
Rm⊥Rmkφ (Rm⊥) ≥ 16a π H 1 + 2 H L 2! (50) with φ as introduced with Eq. (20). We may rewrite this into
Rm⊥Rm∗kφ (Rm⊥) ≥1 (51) with Rm∗kdefined by Rmk= 16a π H 1 + 2 H L 2! Rm∗k. (52)
Figure 18 shows the neutral line, Rm⊥Rm∗kφ (Rm⊥) =1, in
the Rm⊥Rm∗k-plane.
We return now to subharmonic fields B adjusted as de-scribed above to our dynamo box. Within this framework we rediscover the result Eq. (50), or Eq. (51), of the mean-field approach in the double limit L/H → ∞ and a/H → 0. Any deviation of L/H and a/H from this limit leads to higher requirements for dynamo action. In particular, if Rm⊥ is
fixed, higher values of Rm∗kare necessary. For example, for L/H ≤2, a/H → 0 and Rm⊥≤2 the necessary values of
Rm∗kare up to about 10% higher than predicted by the mean-field approach. Figure 18 shows the neutral line for L/H = 2 in the Rm⊥Rm∗k-plane obtained in the mean-field approach
and three such lines derived from subharmonic solutions for finite a/H . Whereas in the mean-field approach dynamo ac-tion seems possible for arbitrary Rm⊥if only Rm∗kis
suffi-ciently large, we see now that it is only possible for not too small Rm⊥. That is, there is not only a critical value of Rm∗k
but also a critical value of Rm⊥so that a dynamo can never
work without exceeding these values. If a/H grows, for any given Rm⊥the requirements to Rm∗kalso grow.
Although the shape of our dynamo box is different from that of the real cylindrical dynamo module and the consid-ered magnetic fields satisfy some kind of periodic bound-ary conditions rather than such which are realistic for this
Fig. 18. Neutral lines for the rectangular dynamo box with
L/H = 2 in the Rm⊥Rm∗k-plane. Line (a) is defined by the
result Rm⊥Rm∗kφ (Rm⊥) = 1 of the mean-field approach. The
other lines are derived on the basis of subharmonic solutions for
L/H =2 and various values of a/H , (b) for a/H = 0.177, (c) for
a/H =0.283 and (d) for a/H = 0.356. The diagram essentially represents results shown in Fig. 6 of Plunian and R¨adler (2002).
module, we may assume that the dependence of the self-excitation condition on the aspect ratio a/H for the real dy-namo module is similar to that observed here. Considering that L/H = 2 and a/H = 0.3 correspond roughly to the real cylindrical dynamo module and that the dynamo works in a regime with Rm⊥, Rmk<2, which implies Rm∗k<0.9, we
may conclude that the marginal value C∗can again be up to 10% higher than predicted by the mean-field approach.
We may further conclude that the neutral line of the ex-perimental device does not need to coincide exactly with an isoline of C in the VCVH-diagram like Figs. 11 or 12.
Us-ing the results represented in Fig. 18, expressUs-ing Rm⊥ and
Rm∗kby V⊥and Vkand putting as in the context of Eq. (21)
again V⊥ = VH and Vk = VC+VH, we have constructed
the neutral lines in VCVH-diagram. For reasons of
compa-rability of these lines we have introduced ˜VC =
√ H /a VC
and ˜VH=
√
H /a VH. Figure 19 shows the neutral lines in a
˜
VCV˜H-diagram. The lines based on the subharmonic analysis
with finite a/H deviate from that obtained in the mean-field approach in the same sense as in Fig. 13 the experimentally determined neutral line deviates from the isolines of C which were obtained in the mean-field approach. That is, this de-viation is understandable as a consequence of the neglect of higher-order derivatives of B in the mean-field approach. 7.4 A kinematic dynamo model with α⊥ varying across
some boundary layer
Quite a few numerical investigations have been carried out with a kinematic dynamo model which deviates from those considered in Sect. 3 by assuming that α⊥ decays from its
value in the interior of the dynamo module across some boundary layer to zero (R¨adler et al., 1999). All induction
Fig. 19. Neutral lines for the rectangular dynamo box with L/H =
2 in the ˜VCV˜H-plane. Both ˜VCand ˜VHin m3/h. The labels (a), (b), (c) and (d) correspond to those in Fig. 18.
Table 3. Marginal values C∗for magnetic fields with different m. The lowest value of C∗for m = 0, which is given here, belongs to field of AS type.
m 0 1 2 3 4
C∗ 8.432 7.276 9.262 11.35 13.54
effects other than the α⊥-effect were again neglected.
In order to explain the distribution of α⊥ and σ , which
was chosen with a view to the real structure of the dynamo module, we define first a small cylinder by r ≤ 0.941R and |z| ≤ 0.458H , and a large cylinder by r ≤ 1.081R and |z| ≤ 0.680H , where r = px2+y2. We assume that α
⊥
is constant inside the small cylinder, decreases in the space between the cylinders linearly in both r and z and vanishes on the surface of the large cylinder and outside it. Consider-ing the large cylinder to be embedded in a sphere as shown in Fig. 3 we further assume that σ is constant inside the large cylinder, is constant and smaller by a factor 100 in the re-maining parts of the sphere and vanishes outside this sphere. We adopt the definition (12) of C with α⊥and η interpreted
as their values inside the small cylinder.
Marginal values C∗of C are given in Table 3. The dynamo has again a non-oscillatory behaviour. The dependence of the growth rates λ on C is depicted in Fig. 20. Some aspects of the structure of the magnetic field with m = 1 are shown in Fig. 21. The field is to a large extent concentrated inside the dynamo module and varies there strongly in z-direction.
8 Concluding remarks
The simple kinematic mean-field theory as explained in Sects. 2 to 5 describes indeed essential features of the Karls-ruhe dynamo experiment. It predicts the structure of the most easily excitable magnetic field and the excitation condition in
0 5 10 15 20 -5 0 5 10 15 C m=1 2 3 4
Fig. 20. The growth rates λ in s−1in dependence on C
its dependence on the rates of flow through the axial and the helical channels of the dynamo module. In agreement with these predictions magnetic fields were observed in the ex-periment which correspond to mean fields of the symmetry type m = 1. As explained above it was clear from the very beginning that the marginal value C∗ of C, which defines the excitation condition, is somewhat underestimated by the simple theory. In Sects. 7.1 to 7.3 a few aspects are discussed which explain why the realistic value C∗may well be up to 30% above the prediction of this theory. Considering these improvements of the theory there is again satisfactory agree-ment between experiagree-ment and theory. It seems even surpris-ing that the experimentally determined region of C∗is only about 10% above the prediction of the simple theory. There is a slight deviation of the predicted shape of the neutral line in the plane of the flow rates VCand VHthrough the two types
of channels from the shape of the line derived from the mea-surements. Again this deviation is understandable with the corrections to the simple theory presented in Sect. 7.3. Of course the preferred orientation of the magnetic fields in the experimental device is, again by reasons already discussed, beyond the scope of the simple theory. In Sect. 6 we have studied the back-reaction of the magnetic field on the fluid motion, more precisely the pressure drop due to the mag-netic field and its influence on the flow rates in the channels of the dynamo module, and developed on this basis a simple model for the dynamo in the nonlinear regime. In this way we gave estimates of the saturation field strengths of the dy-namo, which are again in fair agreement with experimental findings.
Appendix A Steady solution of Eq. (13) in the second– order approximation
Consider Eq. (13), which apply for homogeneous fields B, in the steady case in the second-order approximation, that is, η∇2B0= −(B · ∇)u , ∇ ·B0=0 . (A1) We may put
Fig. 21. Structure of the marginal magnetic field with m = 1. Upper
panel: cylindrical surface with radius 0.46 m, lower panel: mid-plane. Vectors: components tangential to the surface, grey encoded: normal component
a = ∇ × ˜a , ∇ · ˜a = 0 , (A2)
so that
u = −∇2a .˜ (A3)
Then we have ∇2(ηB0−(B · ∇) ˜a) = 0, that is
ηB0−(B · ∇) ˜a = ∇8 and 18 = 0, and can conclude that B0= 1
η(B · ∇) ˜a . (A4)
In the case of the Roberts flow, in which u is given by Eq. (14) we have simply
˜ a = (a
π)
2u . (A5)
For the spin generator flow with u defined by Eq. (22) inside the considered cell and being equal to zero outside we have ˜ a% =0 ˜ aϕ = % 2 a/2 Z % uϕ(%0)d%0+ 1 2% % Z 0 uϕ(%0)%02d%0
Table A1. Some values of xand y
VH[m3/h] 0 25 50 75 100 125 150 175 200 x 0 0.17 0.30 0.39 0.45 0.47 0.48 0.48 0.47 y 0 0.21 0.44 0.68 0.91 1.13 1.31 1.47 1.61 ˜ az= − a/2 Z % uz(%0)ln(%0/%0)%0d%0 −ln(%/%0) % Z 0 uz(%0)%0d%0. (A6)
Appendix B Concerning the determination of α⊥
for the spin generator flow in the second-order approximation
In contrast to our explanations on the second-order approxi-mation in Sect. 4.3 we ignore here no longer the fluid motion outside the considered cell but assume again a flow pattern which is periodic everywhere. We continue to use, however, the cylindrical co-ordinate system %, ϕ, z with the axis % = 0 in the centre of a given cell and consider a Fourier decom-position of the fluid velocity u and of the magnetic fields B and B0with respect to ϕ, that is, a decomposition into modes proportional to exp(imϕ). As for u its part inside the given cell contributes only to modes with m = 0, and that out-side due to the symmetry of the flow pattern only to modes with m = ±4, ±8, · · ·. Since B possesses only modes with m = ±1, the parts of u outside the given cell produce in the second-order approximation no other modes of B0than such with m = ±3, ±5, ±7, · · · . Thus these parts of u produce no contributions to the %, ϕ or z-components of u × B0 inside the given cell other than such with these m. Consequently its x, y and z-components possess only contributions with |m| ≥2. These vanish under averaging over this cell, that is, they do not contribute to u × B0.
These considerations also make clear that in higher than second-order approximations the motion in neighbouring cells may well influence the average of u × B0over the given cell.
Appendix C Relations between local and mean magnetic fields on the axis of the dynamo module
The magnetic probes on the axis of the dynamo module mea-sure the components of the local magnetic field B, which dif-fers from the mean field B by the fluctuations B0. According to the construction of the module the rotational motion of the fluid in the four spin-generators around the axis corresponds to flows away from x = y = 0 in the vicinity of the x-axis
and towards x = y = 0 in the vicinity of the y-axis; see Fig. 2. Assuming that B can be considered as a homoge-neous field and using symmetry arguments we can conclude that Bx0 = −xBx, By0 =yBy, Bz0 =0 , (C1) or Bx= Bx 1 − x , By= By 1 + y , Bz=Bz, (C2)
at the axis of the dynamo module, with positive coefficients x and y depending on VH but not on VC. These relations
have been confirmed by numerical solutions of Eq. (13) for the spin generator flow. Some values of xand y obtained
with these calculations are given in Table 4. Note that B and B, and in particular their directions, can differ markedly.
Acknowledgements. The authors thank Prof. U. M¨uller for many
helpful comments on the manuscript and Dr. M. Sch¨uler for his assistance in preparing a number of figures of this paper.
References
Busse, F. H.: A model of the geodynamo, Geophys. J. R. Astron. Soc., 42, 437–459, 1975.
Busse, F. H.: Magnetohydrodynamics of the Earth’s dynamo, Annu. Rev. Fluid Mech., 10, 435–462, 1978.
Busse, F. H.: Dynamo theory of planetary magnetism and labora-tory experiments, in Evolution of Dynamical Structures in Com-plex Systems, edited by R. Friedrich and A. Wunderlin, vol. 69, pp. 197–207, Springer-Verlag Berlin, 1992.
Busse, F. H., M¨uller, U., Stieglitz, R., and Tilgner, A.: A two–scale homogeneous dynamo: An extended analytical model and an ex-perimental demonstration under development, Magnetohydrody-namics, 32, 235–248, 1996.
Busse, F. H., M¨uller, U., Stieglitz, R., and Tilgner, A.: Sponta-neous generation of magnetic fields in the laboratory, in Evo-lution of Spontaneous Structure in Dissipative Continuous Sys-tems, edited by F. H. Busse and S. C. M¨uller, pp. 546 – 558, Springer, 1998.
Dobler, W. and R¨adler, K.-H.: An integral equation approach to kinematic dynamo models, Geophys. Astrophys. Fluid Dyn., 89, 45 – 74, 1998.
Fuchs, H., R¨adler, K.-H., and Sch¨uler, M.: A numerical approach to dynamically consistent spherical dynamo models, in The Cos-mic Dynamo, (Eds) Krause, F., R¨adler, K.-H., and R¨udiger, G., pp. 129 – 133, Kluwer Academic Publishers Dordrecht Bosten London, 1993.
Gailitis, A.: Conditions of the self–excitation for a laboratory model of the geomagnetic dynamo, Magnitnaya Gidrodinamika, 1967, 45–54, in Russian, 1967.
Krause, F. and R¨adler, K.-H.: Mean–Field Magnetohydrodynam-ics and Dynamo Theory, Akademie–Verlag Berlin and Pergamon Press Oxford, 1980.
M¨uller, U. and Stieglitz, R.: Can the Earth’s magnetic field be simulated in the laboratory?, Naturwissenschaften, 87, 381–390, 2000.
M¨uller, U. and Stieglitz, R.: The Karlsruhe dynamo experiment,
Nonlinear Processes in Geophysics, in print, 2002.
Plunian, F. and R¨adler, K.-H., Subharmonic dynamo action in the Roberts flow, Geophys. Astrophys. Fluid Dyn., in print, 2002. R¨adler, K.-H. and Brandenburg, A.: Contributions to the theory of
the Karlsruhe dynamo experiment, in preparation, 2002. R¨adler, K.-H., Apel, A., Apstein, E., and Rheinhardt, M.:
Contribu-tions to the theory of the planned Karlsruhe dynamo experiment, Tech. rep., Astrophysical Institute Potsdam, 1996.
R¨adler, K.-H., Apstein, E., Rheinhardt, M., and Sch¨uler, M.: Con-tributions to the theory of the planned Karlsruhe dynamo exper-iment – Supplements and Corrigenda, Tech. rep., Astrophysical Institute Potsdam, 1997a.
R¨adler, K.-H., Apstein, E., and Sch¨uler, M.: The alpha–effect in the Karlsruhe dynamo experiment, in Proceedings of the Inter-national Conference ”Transfer Phenomena in Magnetohydrody-namic and Electroconducting Flows” held in Aussois, France, 1997, pp. 9–14, 1997b.
R¨adler, K.-H., Apstein, E., Fuchs, H., and Rheinhardt, M.: Kurzbericht ¨uber Untersuchungen zum Projekt GEODYNAMO: Absch¨atzungen zu den Lorentz–Kr¨aften. Zeitliche Entwicklung eines Magnetfeldes und maximale Feldst¨arke, Tech. rep., Astro-physikalisches Institut Potsdam, 1998a.
R¨adler, K.-H., Apstein, E., Rheinhardt, M., and Sch¨uler, M.: The Karlsruhe dynamo experiment – a mean–field approach, Studia geoph. et geod., 42, 224–231, 1998b.
R¨adler, K.-H., Apstein, E., and Rheinhardt, M.: Kurzbericht ¨uber Untersuchungen zum Projekt GEODYNAMO: Ein modifiziertes Modell des Dynamomoduls. Fremderregte Magnetfelder im un-terkritischen Regime des Dynamos, Tech. rep., Astrophysikalis-ches Institut Potsdam, 1999.
R¨adler, K.-H., Apstein, E., Fuchs, H., and Rheinhardt, M.: Kurzbericht ¨uber Untersuchungen zum Projekt GEODYNAMO: Studien zum Verhalten des Dynamos im nichtlinearen Regime auf der Grundlage eines einfachen Modelles. Absch¨atzungen zum Einfluss eines Magnetfeldes auf den Str¨omungsverlauf in den Spingeneratoren und den α-Koeffizienten, Tech. rep., Astro-physikalisches Institut Potsdam, 2000a.
R¨adler, K.-H., Apstein, E., Fuchs, H., and Rheinhardt, M.: Kurzbericht ¨uber Untersuchungen zum Projekt GEODYNAMO: Stabile und instabile station¨are Zust¨ande des Dynamos im nicht-linearen Regime, Tech. rep., Astrophysikalisches Institut Pots-dam, 2000b.
Roberts, G. O.: Dynamo action of fluid motions with two-dimensional periodicity, Phil. Trans. Roy. Soc. London A, 271, 411–454, 1972.
Soward, A. M.: Fast dynamo action in a steady flow, J. Fluid Mech., 180, 267–295, 1987.
Stieglitz, R. and M¨uller, U.: GEODYNAMO - Eine Versuchsan-lage zum Nachweis des homogenen Dynamoeffektes, Tech. rep., Forschungszentrum Karlsruhe f¨ur Technik und Umwelt, Wis-senschaftliche Berichte FZKA 5716, 1996.
Stieglitz, R. and M¨uller, U.: Experimental demonstration of a ho-mogeneous two-scale dynamo, Phys. Fluids, 13, 561–564, 2001. Tilgner, A.: A kinematic dynamo with a small scale velocity field,
Phys. Lett. A, 226, 75–79, 1996.
Tilgner, A.: Predictions on the behavior of the Karlsruhe dynamo, Acta Astron. et Geophys. Univ. Comenianae, 19, 51–62, 1997. Tilgner, A. and Busse, F.: Subharmonic dynamo action of fluid
mo-tions with two-dimensional periodicity, Proc. R. Soc. London, Ser. A, 448, 237–244, 1995.
